Since this is a convergent alternating series (with terms decreasing in magnitude
and approaching 0), , which will not affect the fourth decimal place.
Then, correct to four decimal places,
BC ONLY†C7. Power Series over Complex Numbers
A complex number is one of the form a + bi, where a and b are real and i^2 = −1.
If we allow complex numbers as replacements for x in power series, we obtain
some interesting results.
Consider, for instance, the series
When x = yi, then (1) becomes
Then
since the series within the parentheses of equation (2) converge respectively to
cos y and sin y. Equation (3) is called Euler’s formula. It follows from (3) that eiπ
= − 1,
and thus that
eiπ + 1 = 0,sometimes referred to as Euler’s magic formula.